Positivity of the generalized translation associated with the \(q\)-Hankel transform
The authors study the positivity of a \(q\)-analogue of the Hankel translation \[ T_x^{\nu}f(y)=\int_0^{\infty}D_{\nu}(x,y,t)f(t)\,dt, \] where \[ D_{\nu}(x,y,z)=\int_0^{\infty}t^{-\nu-1/2}(xt)^{1/2}J_{\nu}(xt)(y)^{1/2}J_{\nu}(yt)(zt)^{1/2}J_{\nu}(zt)\,dt. \] This \(q\)-analogue is defined in terms of the normalized third Jackson \(q\)-Bessel function and its positivity was already posed in [\textit{L.~Dhaouadi, A.~Fitouhi} and \textit{J.~El~Karmel}, JIPAM, J. Inequal. Pure Appl. Math. 7, No. 5, Paper No. 171, 14 p. (2006; Zbl 1232.26025)]. For the positivity Graf's addition formula for the third Jackson \(q\)-Bessel function plays a central role. This addition formula was proved in [\textit{H.~T. Koelink} and \textit{R.~F.~Swarttouw}, J. Approximation Theory 81, No. 2, 260--273 (1995; Zbl 0821.33010)].
scientific article; zbMATH DE number 6005983
scientific article